Global torus blocks in the necklace channel

We continue studying of global conformal blocks on the torus in a special (necklace) channel. Functions of such multi-point blocks are explicitly found under special conditions on the blocks’ conformal dimensions. We have verified that these blocks satisfy the Casimir equations, which were derived in previous studies.


Introduction
In CFT 2 , the main observables are the correlation functions of local operators.The global and local symmetries of the theory impose restrictions on these correlators.In particular, multi-point correlators of primary operators can be decomposed into conformal blocks [1], which are determined only by the symmetries of the theory.Conformal blocks associated with local symmetry algebras such as Virasoro (or extended W N algebras [2,3]) have been studied on the sphere [1,[4][5][6] and on the torus [8][9][10][11].
The global blocks (associated with sl(2, R) algebra) on the sphere were studied in [12][13][14][15].Furthermore, these blocks arise in the large-c limit1 from the Virasoro blocks in different regimes [4,18,19] .On the torus, the study was initiated in [8], where the 1-point block was computed.For N ≥ 2 correlation functions, there are multiple conformal blocks.For instance, N -pt global blocks in the OPE channel [10] were explicitly found to be a product of the (N + 2)-point global block on the sphere and the 1-point torus block.
In this work, we focus on another type of global torus blocks -so-called necklace blocks [20,21].The crucial point is that various mixed channel blocks (where inserting projectors and taking OPE are combined) can be computed using the necklace channel block (see section 2 in [21] for details).In this sense, the necklace block corresponding only to the insertion of projectors and the OPE block which is responsible for pure OPE decompositions generate blocks with an arbitrary channel topology.Nevertheless, functions of necklace blocks are not known in the closed form but it was shown that they are subject of Casimir equations [21].Here, we directly construct the N -point necklace block in terms of a particular (N + 2)-point comb channel block [7] and find the functions of the necklace blocks after imposing special conditions on the conformal dimensions.We also show that these blocks satisfy the Casimir equations mentioned above.
This paper is organized as follows.In Section 2 we introduce the torus conformal blocks in the necklace channel and present a relation between the N -pt necklace channel block and the (N + 2)-pt comb channel block.Section 3 contains examples of necklace block functions.By imposing certain conditions on the comb channel block, we calculate the N -pt torus conformal blocks in the necklace channel.Section 4 is devoted to checking that the found blocks satisfy the Casimir equations.Section 5 summarizes our results and introduces future directions.In Appendix A we list several necklace block functions that supplement the narration in Section 3. A highest weight state is defined as follows

Torus global conformal blocks
and the corresponding Verma module V h is spanned by the descendants states We will often use sl(2, C) states |m, m , h, h = |m, h ⊗ |m , h and associated Verma modules V h, h.The projector on a Verma module V h i takes the form where (2h i ) m = Γ(2h i + m)/Γ(2h i ) is the (rising) Pochhammer symbol.Here the standard conjugation (L −1 ) † = L 1 is assumed and for (2.4) we have where in the first formula the sum is carried out over the chiral part of the spectrum D. Primary operators O h, h(z, z) can be introduced via the operator-state correspondence In addition, the algebra sl(2, R) acting on them can be realized in terms of differential operators with respect to the variable z In what follows, we will also use L

(k)
P to denote the operators (2.7) with respect to the variable z k .Note that the commutation relations for such operators differ from (2.1) by the sign on the right side.
Torus matrix elements.The N -point correlation function of primary operators O i (z i , zi ) with conformal dimensions (h i , hi ) on the torus is given by where Tr H stands for the sum over all Verma modules with weights (h α , hα ), i.e. (2.9) Here we apply L 0 |m, h a = (h α + m)|m, h a to isolate a q-dependence.The N -point function satisfies Ward's identities associated with the u(1) ⊕ u(1) symmetry on the torus The matrix element in the second line of (2.9) can be expressed in terms of the (N + 2)-point correlation function on the sphere with two additional operators with dimensions (h α , hα ).Indeed, using (2.6) the differential realization of the descendant states reads where we exploit the fact that L −1 = ∂.Thus, the expression for the torus matrix element appearing in (2.9) can be cast into the following form (2.12) The illustrative example of the relation above is a computation of the norm |m, m , h α , hα 2 .
Here, we have a 2-pt function O α O α of primary operators with dimensions (h α , hα ) in the second line of (2.12), so where the Pochhammer symbols and factorials come from taking derivatives with respect to z 1 and z 0 , respectively.
The torus block in the necklace channel.The correlation functions (2.8) can be decomposed into conformal blocks in two different ways.The first one corresponds to taking the OPE between operators in (2.8) and the second one to inserting projectors (2.4) on the Verma modules with the set of intermediate dimensions.The former was elaborated in [10] where N -pt OPE blocks were found in a closed form.Here we consider on the latter (necklace) channel, so the correlation function (2.8) can be cast into the following form where F N (q, z|h, h, h α ) is a N -point necklace channel block on the torus which is parameterized by external dimensions of primary operators h j , j = 1, ..., N , intermediate dimensions hi , i = 0, ..., (N − 2) and h α appearing in (2.8) so altogether they are succinctly denoted by a set {h, h, h α }.The notations z, z stand for insertion points of the primary operators.Ch h, h represents as the product of structure constants (2.15) Since the torus matrix element (2.12) can be written through the (N + 2)-pt correlation function on the sphere one can insert (N − 1) projectors which results in expansion into conformal blocks for this correlation function.The corresponding blocks (denoted by G N +2 ) are comb channel blocks [7], so from (2.12) we find where Fig. 1 illustrates the formula (2.16).Notice that concerning the channel topology the necklace N -point block on the torus and the (N + 2)-pt comb channel block differ only in the additional averaging over the conformal family of the operator O α in the first line of (2.16).It is implemented by considering the sum of comb channel blocks with descendant operators ∂ m O α in the first and last place.From the topology standpoint, the limits in the second line of (2.16) can be viewed by gluing endpoints in each comb block together.Finally, in the limit q → 0 the necklace block reads which was studied in details in [21].On the sphere, global conformal blocks in different channels were studied earlier in the literature [7,10,20,22,23].It was found [7] that the comb channel block can be expressed in terms of the so-called comb function defined by For the set of dimensions we have chosen, the comb channel block G N +2 in (2.16) has the form where F K is (2.18), z i,j = z i − z j and Notice that for future needs we have changed the sign in the notation for the first and last parameters of the comb function compared to the definition given in [7].

Explicit block functions
In this section, we obtain necklace block functions using the formula (2.16).Starting from the well-elaborated case of the 1-pt block [8,10,24], we analyze particular examples of 2-and 3-pt blocks and generalize them to N -pt blocks.

1-pt torus block
The global 1-pt torus block was initially considered in the context of large-c limit of Virasoro conformal blocks [8]. 2 Regarding the Casimir approach [10] it was shown that the 1-pt torus block is subjected to the second order differential equation in q.
To calculate the 1-pt torus block in accordance with (2.16), we consider the 3-pt block of primary operators with dimensions (h α , h, h α ) inserted at points (z 0 , z 1 , z 2 ) which has the following form The torus matrix element in the second line of (2.16) was found to be [8] so the first line in (2.16) yields Here the formulas in the last line are obtained by applying the Pfaff transformations [32] to the hypergeometric function in the first line.Notice that for h = 2h α the block (3.3) reduces to which is going to be useful for the further consideration.

N -pt necklace block -preliminaries and assumptions
In order to approach N -pt necklace blocks it is important to mentioned that the key role in computing via (2.16) is played by the dependence on variables z 0 and z N +1 in the comb channel block.For the comb function we can separate z 0 and z N +1 with the help of splitting identities [7] ) so for the (N + 2)-pt comb block (2.19) one has (3.6)One can see that direct implementation of (2.16) to the formula above gives complex expressions.In order to simplify calculations and obtain explicit block functions, we consider particular cases where s 1,2 are given by (2.20).Under this assumption, the double sum in the second line of (3.6) is reduced to the product of finite ones, which contain (s 1 + 1) and (s 2 + 1) terms, respectively.Notice a condition b N −3 = −t 1 , b 0 = −t 2 , t 1,2 = 1, 2, ... can be considered by the same reasons.Despite this, we will still concentrate on the condition (3.7), because in the simplest case t 1,2 = 0 we arrive at G N +2 ∼ z −2hα 01 z −2hα N,N +1 , while for b N −3 = b 0 = 0 one has a more complex structure due to s 1,2 = 0.After setting (3.6), G N +2 reduces to a polynomial (multiplied by the factor z −2hα 01 z −2hα N,N +1 ) in the variables z 02 /z 01 and z N −1,N +1 /z N,N +1 .The computation of the N -pt necklace block under the assumption (3.7) will be illustrated with concrete examples in the next section.We consider obtaining of 2-and 3-pt blocks separately, since the application of (3.7) to the corresponding comb blocks G N +2 gives power functions.In what follows, we slightly change the notation for the necklace block on the torus and move (s 1 , s 2 ) to the superscript.In other words,

2-pt and 3-pt necklace blocks
2-pt block.The 4-pt comb channel block [14,33,34] associated with a 2-pt torus block in the necklace channel reads (3.9) The block G 4 (z 0 , ..., z 3 |h, h, h α ) is parameterized by 4 external dimensions (h α , h 1 , h 2 , h α ) and one intermediate dimension h0 which are denoted by a set (h, h, h α ).Notice that in this case, the formula (3.5) is not applicable explicitly since we only have one summation from a hypergeometric function 2 F 1 in (3.9).Despite this, the general scheme remains the same -setting the first (or the second) parameter of the hypergeometric function to be 0 or a negative integer, one obtains a polynomial instead of an infinite sum In addition to (3.7) for further simplification we set 3 which results in (3.13) The simplest case corresponds to s = 0 where (3.13) reduces to so the second line of (2.16) gives Here, h0 -dependence is excluded by substituting of the condition s = 0. Applying the first line of (2.16) to the formula above we find the 2-pt necklace block (3.16) 3 Without such a requirement the comb channel block (3.9) has a more complex form.Assuming h0 + hα − Analogously, the result for the case s = 1 reads (3.17) There are several comments to make about these formulas.First, one can see that in (3.16) the first factor is the 2-pt correlation function of two primary operators with conformal dimensions h − h α .In Section 4 we prove that this property generalizes to the N -pt necklace block on the torus.Second, the 2-pt block (3.17) factorizes into the product of the s = 0 block (3.16) and a polynomial in the variables q, x.Moreover, for cases s = 2, 3, 4 (see Appendix A.1 for details) one can see that 2 has the following structure where P (s) (q, x) is a polynomial of degree at most 2s in q and x.Polynomials P (s) (q, x) also have a number of properties, for instance, it can be shown that the first of which is essential for the discussion in Section 4. Finally, using ( , we check that in the limit q → 0 the relation (2.17) is satisfied.
3-pt block.In this case the 5-pt comb channel block (2.19) is given by where s 1 and s 2 are defined by (2.20) and F K is the Appell function F 2 [7,32].As previously, it can be presented as a finite sum if s 1,2 are positive integers (3.21)In contrast to the previous case, the condition s 1 = s 2 does not lead to complications so s 1 and s 2 can be independent.For s 1 = s 2 = 0 the double sum above reduces to 1 and using (2.16) we have Few comments are in order.First, one can see that the structure of the first factor resembles one in (3.16) with z 2 replaced by z 3 .Second, G3 is nothing more than a plane 3-pt conformal block of primary operators with dimensions (h 1 −h α , h 2 , h 3 −h α ), located at points (z 1 , z 2 , z 3 ).For s 1,2 = 0 the 3-pt necklace blocks and their properties are listed in Appendix A.2.

N -pt necklace block
For s 1 = s 2 = 0 the computation of the N -pt torus block in the necklace channel mimics one for the 3-pt block described above.In this case, the associated (N + 2)-pt comb channel block is given by (2.19).After setting s 1 = s 2 = 0 in (2.19) and applying (2.16) the N -pt necklace block is found to be where and GN (z|h, h) is the particular N -point comb channel block with a set of external dimensions (h 1 − h α , ..., h N − h α ) and (N − 3) intermediate dimensions ( h1 , ..., hN−3 ) (see Fig. 2).The structure of the block (3.23) remotely resembles one for the N -pt OPE channel block on the torus which was discussed in [10].There, the N -pt OPE block (for any set of conformal dimensions) is factorized into a product of the 1-pt torus block (3.3) and the (N + 2)-pt plane block in some channel which was determined by the OPE's topology of the torus block and could not be a comb channel (see section 4.1.in [10] for details).For the necklace channel blocks on the torus, we have the function (3.24) instead of the 1-pt block and the second factor is the N -pt block in the comb channel with the first and the last dimensions shifted by h α .It is also interesting that if the condition h = 2h α analogous to (3.7) is applied to the 1-pt block we get the function (3.4) which coincides with (3.24) for where the extra terms in the last two equalities appear from shifting the conformal dimensions in the block (see discussion under (3.23)).It is worth emphasizing that structures of this type but with a part of the operators are multiplied by q, are explicitly exuded in the Casimir equations (4.1).Having (4.3), one can easily prove the Ward identity (4.2) for the block (3.23) (4.4) where the last equality is verified explicitly using (3.24).Then, there are the following properties of the function P (0) The equations (4.5) remotely resemble the Ward identities for the conformal block on the plane deformed by the parameter q.Besides, one can find a relation between derivatives with respect to q and z Returning to the equations (4.1) it is necessary to discuss how exactly they reduce to the Casimir equations for the N -pt comb channel block.For the first equation (4.1) the substitution of (4.3) gives us which is satisfied for P (0) (q, z 1 , z N |h α ) defined by (3.24).The second and the last (j = N ) equations in (4.1) after a little bit of algebra involving (4.3), (4.5) and (4.6) can be cast into the form L (1) thus, they are differential Casimir equations associated with external operators of dimensions h 1 − h α and h N − h α .Note that for the examples of 2, 3-pt necklace blocks analyzed above, there are no other Casimir equations that would be related to intermediate block dimensions.
We also check the Casimir equations for the N -pt blocks (3.25) with s 1,2 = 0, 1.For these cases, the Casimir equations for (s 1 + 1)(s 2 + 1) comb blocks G(k,l) N (z|h, h) in (3.25) are used in the proof.The check for larger s 1,2 is straightforward but it is not clear how to manifest that the system (4.1) is satisfied.

Conclusion
In this paper, we studied particular examples of the necklace blocks on the torus corresponding to the condition (3.7).We computed the 2-and 3-pt block functions for the first few s 1,2 which were found to be a product of the particular comb channel block and the factor which carries q dependence.These results were generalized to the N -pt case where such a factorization takes place for the simplest case s 1,2 = 0.For all these necklace block functions, we explicitly checked that they satisfy the Casimir equations (4.1).Namely, it was shown how the given system of equations for the N -pt necklace block is reduced to the Casimir equations for the particular N -pt comb channel block.
Initially, our study was motivated by the search for the exact functions of the necklace blocks (in terms of known special functions) using the condition (3.7).In order to simply analysis one can relax the condition even more and consider degenerate primary operators with h = −l/2, l = 1, 2, ... only.Then, the necklace block (2.16) contains a finite sum over m and is a polynomial in the variable q.It also implies that the comb block reduces to a polynomial in χ i .We anticipate that for such cases it will be possible to find closed expressions for necklace blocks.
The obtained expressions for conformal blocks could help to explore the AdS/CFT correspondence between Wilson lines in the Chern-Simons 3d gravity theory and global torus blocks.Recently, it has been studied for the blocks with bosonic (h = −j, j = 1, 2, ...) degenerate operators [28].Notice that for such operators the condition (3.7) is obviously satisfied.It would also be interesting to generalize these results to N = 1 superblocks by calculating them both explicitly and by analyzing as solutions of the Casimir equations.Another interesting direction is the analysis of these blocks and the Casimir equations (4.1) in the context of the Knizhnik-Zamolodchikov-Bernard equation [35][36][37].

sl( 2 )
representation theory and correlation functions on the torus.We consider CFT 2 with the local sl(2, C) sl(2, R)⊕sl(2, R) symmetry on the torus which is characterized by the modular parameter q.The generators of the holomorphic part are denoted by L m and they satisfy the commutation relations [L m , L n ] = (m − n)L m+n , m, n = 0, ±1.(2.1)For the algebra sl(2, R) we have same relations in terms of generators Lm and [L m , Ln ] = 0.

Figure 1 .
Figure 1.Visualisation of the relation (2.16).The diagram on the left corresponds to the N -pt necklace block.One is represented as a sum of the (N + 2)-pt comb channel blocks with two additional operators ∂ m O α .Red dots are depicted derivatives with respect to z 0 and z N +1 .

Figure 2 .
Figure 2. The N -pt necklace channel block under conditions h α + h0 − h 1 = h α + hN−2 − h N = 0, which are shown by the green dots in the left picture.According to the (3.23), this block factorizes into the product of the function (3.24) (red) and the N -pt comb block with specific dimensions depicted by blue dots.